January 2019 - mathfun@mailman.xmission.com

Re: [math-fun] Could change of base (binary —> ternary) speed up computation?
by Dan Asimov 07 Jan '19

07 Jan '19

There may be answers in this freely downloadable publication of the AMS: <https://www.ams.org/publications/authors/mit-2.pdf>, "Mathematics Into Type" (updated edition). Dan Andy Latto wrote: ----- ... e doesn't have to show up unless you want to figure out what the constant you need to multiply by is. As an aside, how do you deal with sentences like the immediately proceeding one, which start with a mathematical expression, which means they can't start with a capital letter, which looks funny to my eye (even if typeset, so that the e is in math-italics). I usually rephrase; in this case, I might have said "The constant e" or "Euler's e" instead of just "e" to start the sentence. Is there some "style guide for mathematicians", and what does it say about starting a sentence with a formula? -----

1 0

Re: [math-fun] Could change of base (binary —> ternary) speed up computation?
by Dan Asimov 07 Jan '19

07 Jan '19

Mike Speciner wrote: ----- What does c have to do with it? ----- Nothing, as far as I can tell. But a logarithm must have a base. ----- And why do you think a trit would take the same amount of space as a bit? ----- I was just counting storage *locations* for bits vs. trits. —Dan ----- On 06-Jan-19 16:59, Dan Asimov wrote: > Since useful quanputers might be a long ways off, maybe some other stunt > could speed up current computation as we know it. > > If trits could be stored, retrieved, and copied as easily as bits, there > would be less *space* needed by a factor of of log_c(2)/log_c(3) = .6309+, > where c = zeta(3). > > Perhaps someone knowledgeable can estimate how much trits could actually > speed up computation. Surely this has been much studied by now. > > But the practical question is, Is there a practical way to build trit-based > processors? Would it be easier if they could be as large as a room, or larger? -----

4 3

[math-fun] Could change of base (binary —> ternary) speed up computation?
by Dan Asimov 06 Jan '19

06 Jan '19

Since useful quanputers might be a long ways off, maybe some other stunt could speed up current computation as we know it. If trits could be stored, retrieved, and copied as easily as bits, there would be less *space* needed by a factor of of log_c(2)/log_c(3) = .6309+, where c = zeta(3). Perhaps someone knowledgeable can estimate how much trits could actually speed up computation. Surely this has been much studied by now. But the practical question is, Is there a practical way to build trit-based processors? Would it be easier if they could be as large as a room, or larger? —Dan

3 2

[math-fun] Another Klein bottle puzzle
by Dan Asimov 06 Jan '19

06 Jan '19

Think of the Klein bottle K as the square [0,1]^2 with identifications: 1. Top edge identified to bottom edge directly: (x, 0) ~ (x, 1). 2. Left edge identified to right edge in reverse: (0, y) ~ (1, 1-y). The first identification makes the square into a cylinder. The second one identifies the two circular boundaries of the cylinder with each other by reflection about a diameter, reversing orientation. Consider any two simple closed curves in K *equivalent* if one can be continuously deformed to become the other. Puzzle: ------- Find one representative of each equivalence class of simple closed curves on K. —Dan

1 0

Re: [math-fun] An infinite number of mathematicians walk into a bar
by Dan Asimov 04 Jan '19

04 Jan '19

I love this! —Dan Allan Wechsler wrote: ----- An infinite number of mathematicians walk into a bar. First mathematician goes up to the bartender and orders a pint. Second mathematician orders two pints. Third mathematician orders three pints. Fourth mathematician orders four pints. Fifth mathematician orders five pints. Bartender says, "Stop right there! at the rate you're going, you'll owe me a twelfth of a pint." -----

5 4

[math-fun] Another infinite joke...
by William R Somsky 03 Jan '19

03 Jan '19

You're visiting your friend at her apartment, and start discussing mathematical paradoxes. "Suppose," you say, "that you flip the light switch on, then after a second you flip it off. Then, after a further half-second you flip it on again, and after another quarter-second flip it off. Keep doing this on-off-on-off for shorter and shorter intervals, halving their length each time. After two seconds you will have flipped the switch an infinite number of times and you stop. Is the light on or off?" "Oh, I couldn't do that," she says. "But suppose you could," you counter. "But I couldn't!" she insists. "I'd have to call building maintenance for that. That's a super-task!"

1 0

Re: [math-fun] An infinite number of mathematicians walk into a bar
by Dan Asimov 03 Jan '19

03 Jan '19

As I was just about to laugh after reading the punch line, I realized there was nothing to indicate that the m'cians didn't know their limits, and therefore the joke isn't as funny as it might have been, so I didn't. But maybe if some kind of twist were worked in so that they didn't, I would. —Dan ----- An infinite number of mathematicians walk into a bar. First mathematician goes up to the bartender and orders a pint. Second mathematician orders half a pint. Third mathematician orders a quarter pint. Fourth mathematician orders an eighth pint. Fifth mathematician orders a sixteenth pint. Bartender says, "stop right there, here are two pints. The problem with you mathematicians is you need to know your limits." -----

2 1

[math-fun] An infinite number of mathematicians walk into a bar
by Brent Meeker 03 Jan '19

03 Jan '19

An infinite number of mathematicians walk into a bar. First mathematician goes up to the bartender and orders a pint. Second mathematician orders half a pint. Third mathematician orders a quarter pint. Fourth mathematician orders an eighth pint. Fifth mathematician orders a sixteenth pint. Bartender says, "stop right there, here are two pints. The problem with you mathematicians is you need to know your limits." Brent

5 4

Re: [math-fun] Is there a name for the (presumably 100% of) algebraic numbers inexpressible in radicals?
by Dan Asimov 03 Jan '19

03 Jan '19

The goal of Tim Chow's paper — to define "closed form" numbers — reminds me of this one by Maxim Kontsevich and Don Zagier, "Periods": ( https://www.maths.ed.ac.uk/~v1ranick/papers/kontzagi.pdf ). —Dan Fred Lunnon wrote: ----- Chow's paper also at https://arxiv.org/abs/math/9805045 ... -----

1 0

[math-fun] The 2017, 2018, 2019 question
by Ed Pegg Jr 02 Jan '19

02 Jan '19

At reddit, someone posted that 2017, 2018, 2019 are a prime, 2 times a prime, and 3 times a prime. 13, 14, 15 is the first example. https://oeis.org/A074200 and https://oeis.org/A093553 are relevant sequences. In general, we want a number k such that (k+n)/n produces an integer and hopefully a prime for numbers 1 to n. https://oeis.org/A003418 , LCM@@Range[n] gives the smallest values where (k+n)/n is an integer. What if we want the smallest prime factor of (k+n)/n to be bigger than n? Then we need k to be some multiple of the LCM. For n=5 to 40, these multiples seem to be 6, 6, 18, 12, 4, 4, 8, 8, 6, 8, 8, 4, 12, 12, 18, 30, 30, 30, 30, 30, 6, 6, 2, 2, 16, 16, 10, 12, 12, 12, 14, 14, 42, 56, 56, 56 If the multiples are always even, the sequence 41 to 66 continues 42, 42, 112, 112, 112, 112, 126, 126, 18, 32, 32, 32, 12, 12, 12, 12, 12, 12, 120, 120, 6, 6, 6, 18, 18, 18 but I don't know if that's true. Terms for 26 and 27 are both 2. LCM @@ Range[27] is 80313433200. 2 A003418[27]/n + 1 for n for 1 to 27 has smallest prime factors 29, 31, 41, 43, 71, 101, 113 A multiple of 16 is needed for terms 131 to 136. --Ed Pegg Jr

1 1